Why CBR and Comoving coordinates are not absolute reference frames?

Let us say that we know the Hubble parameter now. Why cannot Cosmic Background Radiation and/or the Comoving 3D frame be taken as absolute reference frames (at least far enough from massive bodies, if this is relevant)?

In such frames and continuously correcting via the scale factor, objects to which no force is applied are at rest. This seems to me an absolute inertial reference frame big at least as the observable universe.

Please try to explain me what I do wrong with this reasoning taking into account that my technical knowledge of GR is basically zero.

Edit: for instance who is travelling between the twins of the paradox will be the one changing comoving coordinates. He does so by firing rockets but the same could be deduced on purely kinematic bases.